Statistics & Applications

Inferential

Medium

applied

If $x_1, x_2, x_3, ..., x_n$ are n observations in a sample then which of following is/are TRUE?

(A) The mean $\bar{x}$ has n degree of freedom

(B) The mean $\bar{x}$ has (n-1) degree of freedom

(C) The standard deviation of the sample has (n-1) degree of freedom

(D) The standard deviation of the sample has n degree of freedom

Choose the correct answer from the options given below:

(A) and (C) only

(A) and (D) only

(B) and (C) only

(B) and (D) only

✅ Correct Option: 1

Degree of freedom refers to the number of values that are free to vary in a calculation.

For the mean $\bar{x}$ calculated from $n$ observations $x_1, x_2, x_3, ..., x_n$ :

All $n$ values are completely independent with no restrictions on any of them. Each observation can take any value freely.

Therefore, the mean $\bar{x}$ has $n$ degrees of freedom.

Statement (A) is TRUE.

For the standard deviation of the sample:

The standard deviation formula is $s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}$

The deviations from the mean must satisfy the constraint $\sum(x_i - \bar{x}) = 0$

Once $(n-1)$ deviations are known, the $n$ -th deviation is determined by this constraint and cannot vary freely.

Therefore, the standard deviation has $(n-1)$ degrees of freedom.

Statement (C) is TRUE.

Statements (B) and (D) are FALSE based on the above analysis.

The correct answer is: (A) and (C) only

2025: 30 May Shift 2

If a 99% confidence interval states that the population mean is greater than 100 and less than 400. Then the sample mean and margin of error respectively are:

2025: 14 May Shift 2

The simple random sample consists of six observations: 5, 8, 10, 7, 10, 14. The point estimate of the population mean is:

2026: 31 May Shift 2

Match the LIST-I with LIST-II

LIST-I		LIST-II
A.	The measurable characteristic of a population is called	I. Statistic
B.	The measurable characteristic of a sample is called	II. Sample size
C.	The number of objects in a sample is called	III. Null hypothesis
D.	A tentative assumption about population	IV. Parameter

Choose the correct answer from the options given below:

2025: 22 May Shift 1

Consider the following hypothesis test:

$H_0: \mu \leq 16$

$H_a: \mu > 16$

A sample of 25 provided a sample mean $\bar{x} = 17$ and a sample standard deviation $s = 3.5$ , then the value of the test statistic in $t$ -test is:

2024: 16 May Shift 1

If $95 \%$ confidence interval for the population mean was reported to be $160$ to $170$ and $\sigma=25$ , then size of the sample used in this study is: (Given $\mathrm{Z}_{0.025}=1.96$ )

2025: 16 May Shift 1

A machine has been producing steel tubes of mean inner diameter of 2 cm. A sample of 10 tubes gives an inner diameter of 2.01 cm and a standard deviation of 0.063 cm. Test the hypothesis that the machine is working in the proper order at 5 % level of significance and answer, which of the following statements are correct?

[Given that : t₉(0.05) = 2.262]

(A) The value of the test statistic is t = 0.476.

(B) The null hypothesis is rejected at 5 % level of significance.

(D) The degree of freedom is 10.

Choose the correct answer from the options given below:

Solution

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